Oscillation condition for first order linear dynamic equations on time scales

Abstract

In this paper, we deal with the first-order dynamic equations with
nonmonotone arguments
\begin{equation*}
y^{\Delta }(t)+\underset{i=1}{\overset{m}{\sum }}p_{i}(t)y\left( \tau
_{i}(t)\right) =0,\text{ }t\in \lbrack t_{0},\infty )_{\mathbb{T}}
\end{equation*}
where \(p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}}, \mathbb{R}^{+}\right) ,\) \(\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{T}\right) \) and \(\tau _{i}(t)\leq t,\) \( \lim_{t\rightarrow \infty
}\tau _{i}(t)=\infty \) for \(1\leq i\leq m\). Also, we present a new sufficient condition for the oscillation of delay dynamic equations on time scales. Finally, we give an example illustrating the result.